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4 Mean Values and Thermodynamics
and that the macroscopic thermodynamic (or ensemble average) number N is
defined as
N ≡
N
p N N .
(4.2.8)
4.2.1 Thermodynamics from the Grand Ensemble
The average energy E in the grand ensemble is defined as
E =
N,r N
p N,r N E r N =
1
N,r N
r N e
−ββ r N e
−γ N .
(4.2.9)
Identification of the ensemble average energy E with the thermodynamic internal
energy U then leads, in much the same manner as Eq. (4.1.2a) was obtained from
Eq. (4.1.1), to
U = −
∂ ln
∂β
λ,V
.
(4.2.10)
From the statistical mechanical defining relation U =
N,r N
p N,r N E r N for the
internal energy, we see that the differential of U is given in general by
dU =
N,r N
p N,r N d r N +
N,r N
r N dp N,r N .
(4.2.11)
We shall now examine the second term in this expression in greater detail by
making use of the fact that, for the grand ensemble, the expression p N,r N =
e
−ββ r N λ N // implies that
ln p N,r N = −ββ r N + N ln λ − ln ,
(4.2.12)
from which we can express the energies r N as
r N =
1
β
N ln λ −
1
β
ln −
1
β
ln p N,r N .
Using this result in the second term of Eq. (4.2.11) gives
N,r N
r N dp N,r N = −
1
β
N,r N
[ln p N,r N + ln − N ln λ] dp N,r N